Question

An inspector of flow metering devices used to administer fluid intravenously will perform a hypothesis test to determine whether the mean flow rate is less than the flow rate setting of 207 milliliters per hour with the level of significance of 0.05. Based on prior information, the standard deviation of the flow rate is assumed to be known and equal to 12 milliliters per hour.

a.) Determine the null and alternative hypothesis.

b.) What is the sample size needed to detect the true mean of 212 milliliters per hour with probability of 0.9?

c.) What assumptions are needed to perform this hypothesis test?

d.) Use the sample size in part b to find the critical region and specify your conclusion for the hypothesis test if the sample mean is 204 milliliters per hour.

e.) Use the sample size in part b to find the P-value for this test and specify your conclusion with the sample mean of 204 milliliters per hour.

Answer 1

a.
Given data,
population mean is 207
population standard deviation is 12
level of significance =0.05
true mean = 212
probability =0.9
type 2 error = 1-power = 1-0.9 =0.1
beta =0.1
b.
sample size = ((population standard deviation^2)*(Zalpha +Zbeta)^2)/(population mean - true mean )^2
sample size = ((12^2)*(Z0.05 +Z0.10)^2)/(212-207)^2
sample size = (144*(1.645 +1.28)^2)/(25) =49.28 = 50
c.
standard normal distribution and population standard deviation is known
d.
Given that,
population mean(u)=207
standard deviation, σ =12
sample mean, x =204
number (n)=50
null, Ho: μ=207
alternate, H1: μ<207
level of significance, α = 0.05
from standard normal table,left tailed z α/2 =1.645
since our test is left-tailed
reject Ho, if zo < -1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 204-207/(12/sqrt(50)
zo = -1.768
| zo | = 1.768
critical value
the value of |z α| at los 5% is 1.645
we got |zo| =1.768 & | z α | = 1.645
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : left tail - ha : ( p < -1.768 ) = 0.039
hence value of p0.05 > 0.039, here we reject Ho
ANSWERS
---------------
null, Ho: μ=207
alternate, H1: μ<207
test statistic: -1.768
critical value: -1.645
decision: reject Ho
e.
p-value: 0.039
we have enough evidence to support the claim that whether the mean flow rate is less than the flow rate setting of 207 millimeter per hour.